Assessment of high‐order IMEX methods for incompressible flow

• Published in: International journal for numerical methods in fluids 2023-06, Vol.95 (6), p.954-978
• Main Authors: Guesmi, Montadhar, Grotteschi, Martina, Stiller, Jörg
• Format: Article
• Language: English
• Subjects: Boundary conditions; Competitiveness; discontinuous Galerkin; Fluid flow; Galerkin method; IMEX Runge‐Kutta methods; Incompressible flow; Laminar flow; Methods; spectral deferred correction; spectral element method; Time dependence; Turbulence; Viscosity
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5177

Summary This paper investigates the competitiveness of semi‐implicit Runge‐Kutta (RK) and spectral deferred correction (SDC) time‐integration methods up to order six for incompressible Navier‐Stokes problems in conjunction with a high‐order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight‐forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time‐dependent boundary conditions, two third‐order RK methods are identified that perform well in all test cases and clearly surpass all second‐order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca 10−5$$ 1{0}^{-5} $$. This paper investigates semi‐implicit Runge‐Kutta and spectral deferred correction methods up to order six for incompressible flow problems. A novel approach based on partitioned Runge‐Kutta methods is proposed for embedding the projection scheme and to achieve a consistent treatment of nonlinear viscosity. Numerical experiments including laminar flow, variable viscosity and transition to turbulence demonstrate the accuracy, convergence and computational efficiency of the proposed methods and their superiority over second‐order methods.